If a, b, c are in

a, b, c, d are in G.P.

Therefore,

bc = ad … (1)

b² = ac … (2)

c² = bd … (3)

To prove: (a² + b²), (b² + c²), (c² + d²) are in G.P, we need to prove that:

(a² + b²) (c² + d²) = (b² + c²)² {deduced using GM relation}

∴ RHS = (b² + c²)² = b⁴ + c⁴ + 2b²c²

= a²c² + b²d² + a²d² + b²c² {using equation 2 and 3}

⁼ c²(a² + b²) + d²(a² + b²)

= (a² + b²) (c² + d²) = LHS

∴ (a² + b²), (b² + c²), (c² + d²) are in G.P

Hence proved.